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Typical Quantum States of the Universe are Observationally Indistinguishable

Eddy Keming Chen, Roderich Tumulka

Abstract

We establish three impossibility results regarding our knowledge of the quantum state of the universe. Suppose the universal quantum state is a typical unit vector in a high-dimensional subspace $\mathscr{H}_0$ of Hilbert space $\mathscr{H}$, such as the low-entropy subspace defined by the Past Hypothesis. We show that: (1) Any particular observation is incapable of identifying the universal state vector in $\mathscr{H}_0$ or substantially reducing the set of possibilities. In other words, the overwhelming majority of possible state vectors are observationally indistinguishable from each other. (2) For any reasonably probable measurement outcome and for most pairs of vectors in $\mathscr{H}_0$, that outcome will not appreciably favor one vector over the other. (3) Bayesian updating on any measurement result, unless it is extraordinarily improbable, has a negligible effect on the initial uniform probability distribution over the states in $\mathscr{H}_0$. These findings represent the most stringent epistemic constraints known for a quantum universe and are derived from a typicality theorem in quantum statistical mechanics. We close by considering how theoretical considerations beyond empirical evidence might inform our understanding of this fact and our knowledge of the universal quantum state.

Typical Quantum States of the Universe are Observationally Indistinguishable

Abstract

We establish three impossibility results regarding our knowledge of the quantum state of the universe. Suppose the universal quantum state is a typical unit vector in a high-dimensional subspace of Hilbert space , such as the low-entropy subspace defined by the Past Hypothesis. We show that: (1) Any particular observation is incapable of identifying the universal state vector in or substantially reducing the set of possibilities. In other words, the overwhelming majority of possible state vectors are observationally indistinguishable from each other. (2) For any reasonably probable measurement outcome and for most pairs of vectors in , that outcome will not appreciably favor one vector over the other. (3) Bayesian updating on any measurement result, unless it is extraordinarily improbable, has a negligible effect on the initial uniform probability distribution over the states in . These findings represent the most stringent epistemic constraints known for a quantum universe and are derived from a typicality theorem in quantum statistical mechanics. We close by considering how theoretical considerations beyond empirical evidence might inform our understanding of this fact and our knowledge of the universal quantum state.

Paper Structure

This paper contains 18 sections, 5 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Distribution Typicality
  3. The Key Theorem
  4. Main Motivation: The Past Hypothesis
  5. Observation Typicality
  6. General Distinctions
  7. Mathematical Descriptions
  8. Not Reliably Distinguishable
  9. Not Even Unreliably Distinguishable
  10. A Bayesian Analysis
  11. Physical Consequences: First Argument for OI
  12. Physical Consequences: Second Argument for OI
  13. Remarks
  14. Experiment Framing vs $|\Psi_t|^2$ Framing
  15. Other Kinds of Indistinguishability
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\mathscr{H}_{0}$ be an arbitrary subspace with finite dimension $d_{0}$ in $\mathscr{H}$, let the density matrix $\rho_{0}=P_{0}/d_{0}$ be the normalized projection to $\mathscr{H}_{0}$, and let $B$ be any self-adjoint operator on $\mathscr{H}$. Then for every $\varepsilon>0$, for $(1-\varepsil where the operator norm $\|B\|$ means the largest absolute eigenvalue of $B$. Moreover, for every P

Theorems & Definitions (12)

  • Theorem 1: Distribution Typicality
  • Example 1
  • Example 2
  • Definition 1
  • Example 3
  • Corollary 1
  • Definition 2
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • ...and 2 more